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Deep learning methods for stochastic Galerkin approximations of elliptic random PDEs

Fabio Musco, Andrea Barth

TL;DR

This work tackles stochastic elliptic PDEs with random forcing and diffusion that may vanish or become unbounded, by replacing traditional stochastic Galerkin solvers with deep learning surrogates. It introduces S-GalerkinNet (strong-form residual) and S-RitzNet (weighted weak-form Ritz energy) to approximate the PC spectral coefficients of the stochastic Galerkin expansion, with the latter offering weaker regularity requirements and applicability to log-Gaussian fields. A theoretical key contribution is a Ritz-energy functional with a unique minimizer, guaranteeing a well-posed stochastic Galerkin approximation in the weighted setting. Empirically, the methods demonstrate enhanced training efficiency and scalability on high-dimensional, fully-coupled SG systems, solving challenging problems (including log-normal-type coefficients) on a single GPU within minutes, thus broadening practical applicability of intrusive uncertainty quantification techniques.

Abstract

This work considers stochastic Galerkin approximations of linear elliptic partial differential equations (PDEs) with stochastic forcing terms and stochastic diffusion coefficients, that cannot be bounded uniformly away from zero and infinity. A traditional numerical method for solving the resulting high-dimensional coupled system of PDEs is replaced by deep learning techniques. In order to achieve this, physics-informed neural networks (PINNs), which typically operate on the strong residual of the PDE and can therefore be applied in a wide range of settings, are considered. As a second approach, the Deep Ritz method, which is a neural network that minimizes the Ritz energy functional to find the weak solution, is employed. While the second approach only works in special cases, it overcomes the necessity of testing in variational problems while maintaining mathematical rigor and ensuring the existence of a unique solution. Furthermore, the residual is of a lower differentiation order, reducing the training cost considerably. The efficiency of the method is demonstrated on several model problems.

Deep learning methods for stochastic Galerkin approximations of elliptic random PDEs

TL;DR

This work tackles stochastic elliptic PDEs with random forcing and diffusion that may vanish or become unbounded, by replacing traditional stochastic Galerkin solvers with deep learning surrogates. It introduces S-GalerkinNet (strong-form residual) and S-RitzNet (weighted weak-form Ritz energy) to approximate the PC spectral coefficients of the stochastic Galerkin expansion, with the latter offering weaker regularity requirements and applicability to log-Gaussian fields. A theoretical key contribution is a Ritz-energy functional with a unique minimizer, guaranteeing a well-posed stochastic Galerkin approximation in the weighted setting. Empirically, the methods demonstrate enhanced training efficiency and scalability on high-dimensional, fully-coupled SG systems, solving challenging problems (including log-normal-type coefficients) on a single GPU within minutes, thus broadening practical applicability of intrusive uncertainty quantification techniques.

Abstract

This work considers stochastic Galerkin approximations of linear elliptic partial differential equations (PDEs) with stochastic forcing terms and stochastic diffusion coefficients, that cannot be bounded uniformly away from zero and infinity. A traditional numerical method for solving the resulting high-dimensional coupled system of PDEs is replaced by deep learning techniques. In order to achieve this, physics-informed neural networks (PINNs), which typically operate on the strong residual of the PDE and can therefore be applied in a wide range of settings, are considered. As a second approach, the Deep Ritz method, which is a neural network that minimizes the Ritz energy functional to find the weak solution, is employed. While the second approach only works in special cases, it overcomes the necessity of testing in variational problems while maintaining mathematical rigor and ensuring the existence of a unique solution. Furthermore, the residual is of a lower differentiation order, reducing the training cost considerably. The efficiency of the method is demonstrated on several model problems.
Paper Structure (10 sections, 4 theorems, 86 equations, 1 figure)

This paper contains 10 sections, 4 theorems, 86 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Contribution of the paper
  3. Elliptic random partial differential equation
  4. Spectral expansion methods
  5. Generalized Polynomial Chaos Expansion
  6. Stochastic Galerkin Method
  7. Computational form of the stochastic Galerkin systems
  8. Deep learning approach
  9. Definition and notation
  10. Neural network architecture and loss function

Key Result

Theorem 3.3

Let $Y = (Y_1, \dots , Y_N)$, for $N \in \mathbb{N}$, be the random variable defined in Definition Def:UnderlyingRVs_and_Distribution, such that the sequence $(Y_n)_{n = 1, \dots , N}$ satisfies Assumption Assumption:RVsReq_gPC_MomentsAndDeterminateMeasure, and let $(p_{n,i})_{i \in \mathbb{N}_0}$,

Figures (1)

  • Figure 1: Architecture of S-GalerkinNet and S-RitzNet.

Theorems & Definitions (16)

  • Remark 2.6
  • Definition 3.1
  • Theorem 3.3
  • Definition 3.4
  • Example 3.5
  • Remark 3.6
  • Definition 3.7
  • Remark 3.9
  • Theorem 3.13
  • proof
  • ...and 6 more